European Mathematical Society - t. g. faticoni
https://euro-math-soc.eu/author/t-g-faticoni
enDirect Sum Decompositions of Torsion-Free Finite Rank Groups
https://euro-math-soc.eu/review/direct-sum-decompositions-torsion-free-finite-rank-groups
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This monograph is devoted to direct sum decompositions of reduced torsion free finite rank (rtffr) Abelian groups. Any Abelian group G of finite rank has an indecomposable decomposition, hence one can study questions related to the uniqueness of indecomposable decompositions of G. The book offers a technique that passes these problems to study the factor of End(G) modulo its nilradical N(End(G)). Let us denote this ring by E(G). The book provides a lot of interesting results not included in other books. The first chapter contains some preliminaries. The second chapter explains some motivation. The Krull-Schmidt-Azumaya and the Baer-Kulikov-Kaplansky theorems are explained as examples of good behaviour. On the other hand, the Corner result and other constructions are mentioned to show almost arbitrarily bad behaviour of direct-sum decompositions of rtffr Abelian groups. The notion of quasi-isomorphism and local isomorphism is introduced and the Jόnsson theorem shows that one gets much better behaviour when considering this problem up to a quasi-isomorphism. Chapter 3 explains how the local isomorphism classes of finitely generated projective modules over a semiprime ring having its additive group rtffr are translated into isomorphism classes of finitely generated projectives over a different ring. The next chapter gives results when some commutativity conditions in End(G) are satisfied. </p>
<p>The fifth chapter investigates what can be said about the number of isomorphism classes of groups locally isomorphic to a strongly indecomposable rtffr group with E(G) being a commutative domain. Chapter 6 studies the Baer splitting property. In chapter 8, the author studies Gabriel filters, in particular the filter of divisibility and its relation to the quasi-splitting of some exact sequences. The last chapter returns to E-properties and possible values of homological dimensions of G over End(G) are discussed. The reader of this book is supposed to be rather advanced in Abelian groups. The exposition is almost self-contained, with just a few results going far beyond the scope of the book (for example results from analytic number theory) and which are stated without proofs. A lot of examples illustrate the theory. The author provides a number of exercises and suggestions for further research at the end of each chapter.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">ppr</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/t-g-faticoni" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">t. g. faticoni</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/chapman-hallcrc-boca-raton-pure-and-applied-mathematics-vol-285" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">chapman & hall/crc, boca raton: pure and applied mathematics, vol. 285</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2007</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-58488-726-3</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 89.96</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/20-group-theory-and-generalizations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">20 Group theory and generalizations</a></li></ul></span>Sat, 01 Oct 2011 11:28:50 +0000Anonymous39859 at https://euro-math-soc.eu